Concerning asymptotic expansions of the distribution of self-normalized sums, Chung [1]} was the first to prove an Edgeworth expansion for the Student \(t\) -statistic. Bhattacharya and Ghosh [2]} provided an approach of Edgeworth expansion including self-normalized sums, relaxing Chung's moment conditions. The moment and smoothness assumptions of [2]} were further weakened in [4]}, [5]}. [6]} attained an expansion for the \(t\) -statistic under minimal moment conditions. One-term Edgeworth expansions were treated in [7]}, [8]}, [9]}, [10]}, [11]}, [12]}. Recently, [13]} used the first two expansion terms to derive a sharper bound in a Berry–Esseen type result. Although not explicitly stated for self-normalized sums, Edgeworth expansions for densities of more general statistics are given in [14]}.
